مقادیر b و a را چنان تعیین کنید که تابع زیر در نقطهٔ $x = - 1$ پیوسته باشد.
$f(x) = \left\{ \begin{gathered}
- 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x \lt - 1 \hfill \\
ax + b\,\,\,\,\,\,\,\,x = - 1 \hfill \\
{x^2} - b\,\,\,\,\,\,\,\,x \gt - 1 \hfill \\
\end{gathered} \right.$
نمایش پاسخ
مقدار تابع = حد چپ = حد راست
$f( - 1) = a( - 1) + b = - a + b$
$\mathop {\lim }\limits_{x \to - {1^ - }} f(x) = \mathop {\lim }\limits_{x \to - {1^ - }} ( - 1) = - 1$
$\mathop {\lim }\limits_{x \to - {1^ + }} f(x) = \mathop {\lim }\limits_{x \to - {1^ + }} {x^2} - b = {( - 1)^2} - b = 1 - b$
$\left\{ \begin{gathered}
1 - b = - 1 \to b = 2 \hfill \\
- a + b = - 1 \to - a + 2 = - 1 \to a = 3 \hfill \\
\end{gathered} \right.$